Computational investigation of stochastic Zika virus optimal control model using Legendre spectral method

This study presents a computational investigation of a stochastic Zika virus along with optimal control model using the Legendre spectral collocation method (LSCM). By accumulation of stochasticity into the model through the proposed stochastic differential equations, we appropriating the random fluctuations essential in the progression and disease transmission. The stability, convergence and accuracy properties of the LSCM are conscientiously analyzed and also demonstrating its strength for solving the complex epidemiological models. Moreover, the study evaluates the various control strategies, such as treatment, prevention and treatment pesticide control, and identifies optimal combinations that the intervention costs and also minimize the proposed infection rates. The basic properties of the given model, such as the reproduction number, were determined with and without the presence of the control strategies. For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_0<0$$\end{document}R0<0, the model satisfies the disease-free equilibrium, in this case the disease die out after some time, while for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_0>1$$\end{document}R0>1, then endemic equilibrium is satisfied, in this case the disease spread in the population at higher scale. The fundamental findings acknowledge the significant impact of stochastic phonemes on the robustness and effectiveness of control strategies that accelerating the need for cost-effective and multi-faceted approaches. In last the results provide the valuable insights for public health department to enabling more impressive mitigation of Zika virus outbreaks and management in real-world scenarios.

the population.Mathematical models can help to regulate the threshold of reproduction number, which bring the information on how long the proposed infection finish.Numerous researchers, including those who have studied the transmission of ZIKV, have investigated mosquito-borne infections using mathematical models [15][16][17] .The elements influencing ZIKV's spread can be found using mathematical models.Effective disease control is a top priority for the WHO, and using optimal time control can provide valuable theoretical insights for disease prevention and treatment.The insights gained from optimal control modeling can inform decision-making and help health care providers plan and deliver effective disease management and treatment.Optimal time control has been applied in numerous studies of mosquito-borne diseases 18,19 .
The ZIKV has unnatural significant public health challenges since its severe congenital effects, emergence and its primarily due to its rapid spread.This has constrained the development of computational and mathematical models to alleviate and understand its spread.However, the optimal control theory has been widely applied to the epidemiological models to construct the strategies for controlling such infectious diseases and its spread in the world.Such models typically consolidate various control measures such as treatment, vaccination and public health interventions to control the impact and spread of the disease.
The deterministic vs. stochastic models In the context of infectious disease modeling, both of the above models approaches have been employed in the field of epidemiology.Deterministic models use initial conditions and fixed parameters to predict the behavior of the system.In the literature there are many researcher studies the deterministic models for different diseases [20][21][22] .Similarly, the deterministic models with fractional derivatives were proposed by 23,24 .However, the deterministic phenomena often fail to capture the inherent randomness in the spread of the diseases, especially when the disease is at an early stage or when the number of infected individuals is small.
Stochastic models, on the other hand, incorporate random variables and processes to better represent the unpredictable nature of disease transmission [25][26][27] .Stochastic models are particularly useful for capturing the variability and uncertainty in the spread of diseases 28,29 .For instance, to demonstrated the stochasticity importance in the context of mosquito-borne diseases, highlighting how stochastic models can provide more realistic predictions as compared to deterministic models 30 .
In the present research study, we focus the LSCM that is one of a powerful numerical technique, that has been used to solve such optimal control problems in field of epidemiology.Moreover, the present technique also offers several advantages, including the high accuracy, fast convergence and ability to handle such complex boundary conditions.
The stochastic Optimal Control of ZIKV contributes to the existing literature by investigating a stochastic optimal control model for the ZIKV using the LSCM.This approach combines the strengths of spectral methods and stochastic modeling to address the complexities corresponding with the spread of ZIKV.Previous studies have explored various aspects of ZIKV control, including vaccination strategies 14,17 , vector control 3,4 , and public health interventions 10,11 .However, the incorporation of stochastic elements and the use of the LSCM provide a novel perspective on this problem.
In this article, we provide a novel model that use optimal control to investigate how prevention, pesticides and mass treatment impact the potential of the ZIKV to spread.We want to reduce the number of hosts and vectors that are infected with ZIKV while maximizing the effectiveness of insecticides, prevention and mass treatment.
Moreover, integration of the stochastic modeling along with the LSCM represents a significant advancement in the field of the proposed epidemiological modeling.The LSCM approach also offers the robust structure for evaluating control strategies and designing for ZIKV, computing for the uncertainties in the disease transmission.The present research work builds on the actual body of the knowledge which demonstrating the possible of the LSCM to control of infectious diseases and improve our understanding.However, the future research can further be analyze the application of the proposed methods to other infectious diseases and clarify the complex models to consolidate the additional complexities and real-world data.

Model construction
The stochastic mathematical model in the form of ordinary differential equation has been formed in this section.Bonyah 31 developed and examined the SIR epidemiological model, as follow: Examining the stochastic phenomena of the suggested ZIKV model is the aim of our study.When opposed to the deterministic technique, the stochastic approach is thought to be more practical, particularly when modeling short-term disease data such as ZIKV and the flu.Stochastic models are better suited for precisely simulating the phenomena of interest.Hence, we have developed a stochastic system based on the equations outlined in Eq. ( 1): The study explores the dynamics of ZIKV transmission to humans using a susceptible; infected; recovered model.The human population at time t, denoted by N H , is divided in this model into three categories: susceptible people (1) S H (t) , infected persons I H (t) , and those who have recovered from Zika R H (t) .The population vectors at time t consists of two sub-populations of mosquitoes, N V (t) : susceptible mosquitoes S V (t) and infected insects I V (t) .β 1 reflects the ZIKV transmission rate from mosquitoes to humans, whereas β 2 represents the transmission rate from humans to mosquitoes.Additionally, the model takes into account the following: the rate of recruitment into the susceptible vector population V , the rate of recruitment into the susceptible human population H , the rates of natural death for both hosts and vectors µ H and µ V respectively, the rate of recovery from treatment υ 1 , and the average infectious period for humans ς .The research examines three limited and Lebesgue integrable control functions: pesticide control µ 3 , treatment control µ 2 , and prevention control µ 1 .All infectious diseases exhibit randomness in their transmission, which can be modeled by introducing white noise into deterministic system equations.In this study, we utilized this approach to obtain a stochastic counterpart of the ZIKV model in Eq. (1).In the case of the Zika virus, various factors can contribute to the variability of the source term.For instance, seasonal variations in mosquito populations, climate conditions affecting mosquito breeding and activity, human travel patterns, and effectiveness of public health interventions can all impact the rate at which new infections occur.Additionally, the variable source term may account for changes in the number of susceptible individuals due to factors like immunization campaigns, migration, or changes in human behavior in response to public health messages.
We employed the LSCM to numerically solve the ZIKV model, which has been previously used for differential and integral systems.Additionally, various authors have used this method to solve different SIR models [32][33][34] .
This research paper is structured as follows: in the next section, we review the Legendre polynomials and spectral method.The basic reproduction number is briefly covered in "Method Description", while "Basic Reproduction Number (R0)" presents the stability analysis.We go over the numerical results in section Stability and Convergence of the Spectral Method" and the last part discusses conclusion.

Method description
Legendre polynomials, which have been well investigated in earlier works [35][36][37] , must be introduced before exploring the Legendre spectral collocation method.Q n (ξ ) represents the n th order Legendre polynomial.Let us consider a function v(ξ ) that is defined as follows on [−1, 1]: The unknown Legendre coefficients are represented by v(ξ i ) , and the collocation points are considered to satisfy the condition −1 = ξ 0 < ξ 1 < ... < ξ n = 1 .Where in genral: .
To model Eq. ( 2) using the "Legendre-Gauss iteration" with a weight function, we utilized Legendre-Gauss Lobatto points {τ j } N j=0 , .To numerically approximate the equations, we took the integral of both sides of Eq. ( 2) over the interval [0, t], resulting in: Keeping in mind the initial values S H (0), I H (0), R H (0), S V (0) , and I V (0) for each of the corresponding classes.We make the transformation u = t 2 (τ + 1) , to explore the Legendre spectral collocation technique (LSCM) on [-1, 1].Equation ( 5) may thus be expressed as follow: www.nature.com/scientificreports/next we covert this equation to semi-discretized spectral form as: where the weight function for deterministic part is: Equation ( 5) is currently being employed to approximate S H (t) , I H (t) , R H (t) , S V (t) and I V (t) using the Legendre polynomial.
The Legendre coefficients S Hm , I Hm , R Hm , S Vm , and I Vm correspond to the functions of each class.Utilizing the aforementioned approximation, we can simplify Eq. ( 7) to a more manageable form as follows: To make the calculation simple we have made the substitution η l = t 2 (τ l + 1) .Hence, the system of equations Eq. ( 9) will have " 5N + 5 " unknowns and 5N nonlinear algebraic equations.The five initial conditions are used to do for the answer: S Vm P m (t), N m=0 So Equations (9 and 10) yield a system containing (5N + 5) nonlinear algebraic equations and and the same number of unknowns S Hm , I Hm , R Hm , S Vm , I Vm for m=0,1,...,N.By substituting the unknown values into Eq.( 8), a solution approximation for the stochastic SIR model may be produced.

Basic reproduction number ( R 0 )
Equation ( 1) provide the dynamics of ZIKV disease,in which the two infectious classes are, By employing the next-generation operator approach,the matrix F stands for new generated infections while the transition terms are taken in matrix V , thus we have The basic reproductive number, R 0 is given to us by the spectral radius of the matrix FV −1 , and hence:

Stability and convergence of the spectral method
We first discuss the following stability, accuracy and convergence analysis for the proposed LSCM. 1.For stability analysis, we first investigate the eigenvalues of the linear operator L in a matrix representation.For this, we have assume a differential equation in the form: To expanding u(x, t) in terms of Legendre polynomials P n (x): Apply the operator L to u N is: Next we construct a matrix A where: For the stability analysis we requires the eigenvalues i of a matrix A that satisfy: This confirm that the proposed system stable and also with time the errors do not grow exponentially that maintaining the stability of the numerical solution using LSCM.
2. Accuracy: The accuracy of the LSCM is due to its exponential convergence properties.Therefore, for a function f (x) with an expansion: the error E N when approximating the given function f with the first N terms is: For the sufficiently smooth function, coefficients a n exponentially decay that is: where C and β are constants.This leads to an exponentially decreasing error: a n (t)P n (x).
a n (t)LP n (x).
where α is related to smoothness of the function f (x).
3. Convergence Analysis: Convergence of the LSCM is to determined by the decay of the proposed spectral coefficients.Therefore, for an analytic function f (x): the coefficients a n decay as: where the parameter γ totaly depends on a distance to the nearest singularity of f in the complex plane.The given decay gives guarantee to the rapid convergence.Similarly, error bound derivation for a proposed function f along with m continuous derivatives, the error E N take the following form: Now using the orthogonality of Parseval's theorem and Legendre polynomials: For an analytic function, |a n | ≤ Ce −βn , so: Evaluating the geometric series: Thus: where . This confirms that the exponential convergence of the proposed LSCM which highlighting its efficiency and accuracy.
As described in Eq. ( 2), we examine the stability analysis of the stochastic ZIKV model system in this section.Remember that the "next generation approach" (NGM) states the following, as expounded above: Theorem 1 The infected class defined in system Eq.(1) decreases inside the system and reaches an infectious-free equilibrium when R 0 ≤ 1, then " S * H (t), , is attained, where Proof The system defined in Eq. ( 1) may be persuade by S * H (t), I * H (t), R * H (t), S * V (t) and I * V (t) The equations in the system may be solved if we think of E* as the endemic equilibrium of Eq. ( 1): Vol:.( 1234567890) , While the fourth equation of the system produces S * V (t) = � V µ V +υ 2 µ 3 , the third equation of the proposed system given in Eq. ( 13) follows that R * H (t) = 0 .The following is obtained by inserting these com- puted values into the system's first equation: S * H (t) = � H µ H At last, a state of equilibrium has been attained, " (S * H (t), The relevant epidemic equilibrium is then found by using Maple-13 if I * H (t) > 0,I * V (t) > 0.

Lemma 2
The entire region D remains a positive invariant set with in the context of the stochastic system provided in Eq. (2).
Proof As for our prop os e d mo del we have des cr ib e d t he p opu l at ion as under (2), we obtain Integrating and solving we have providing that, the region is positively invariant.

Lemma 3
The system defined in Eqs.(1) with variables S H (t), I H (t), R H (t), S V (t), I V (t) ∈ R 5 , for every initial condition, displays the following characteristics: and similarly certainly for every class.
Definition 4 Human infected individuals I H as well as the vector infected population I V , become extinct if and only if lim t→∞ I(t) = 0 in Eq. (2).
the stochastic systems represented by Eq. (2) exponentially approaches to zero.Consequently for R 0 >1 the infected class does not vanish and remain present in Eq. (2),where V Proof Assume for the moment that the initial conditions of the stochastic system given by Eq. ( 2) are true and that a solution , S H (t), I H (t), R H (t), S V (t), I V (t) exists.that satisfies these conditions.Further I = I H + I V The following form might then be obtained using the Itô's formula: Eq. ( 14) need to be integrated with limits from 0 to t taking the following shape: We discuss two cases, if  16) by a positive t, then With the application of Lemma 5.4, and lim t→∞ the following result is obtained from Eq. ( 17): The second case if , then from Eq. ( 15) we have The following equation is the result of division of Eq. ( 18)'s both sides by positive t i.e: Again with the application of Lemma 5.4, and lim t→∞ the following result is obtained from Eq. ( 19): This shows that lim t→∞ I(t) = 0.
Theorem 6 Equation (2) indicates the existence of the contaminating population, I H and I V , if the reproduction number R0 is greater than 1.For evidence, see 39 .

Numerical discussions
The numerical test problems are included in this section.Both the stochastic system Eq.( 2) and the deterministic system Eq.( 1) obtain and describe the numerical findings.The numerical results for the specified models are found using the Legendre spectral collocation method.The outcomes are displayed in Fig. 1 through 10.On a home computer, related computations are performed using the programs Matlab and Maple.In Fig. 1, it was simple to set the right parameter values.We used the following parameter values in the context of the deterministic system given by Eq. ( 1): Wit h these parameter values, the calculation produced a reproduction number R 0 = 0.813587 < 1 .In this case, the application of Theorem 1, suggest that disease die out and hence we have only S H and S V classes refer to as E 0 (3, 0, 0, 2, 0) , where the vector population H = 3 and the human population V = 2.
Similarly, in Fig. 2 we increase the contact parameters β 1 = 0.75 and β 2 = 0.75, " and with all other parameter values consistent with those shown in Fig. 1.The reproduction number in this case is R 0 = 1.675 > 1 , and hence disease exist in the population as Fig. 2 illustrates.These results are consistent with Theorem 1 claims.
In Fig .3 , for t he sto chast ic system E q. (2 ), we s ele c te d p arameter va lues σ S H dW(t) ln σ S H (t)dW(t) predicts a decrease in the number of infected persons in the vector and human populations inside the system, in accordance with the ideas presented in Theorem 5, with R0 = 0.6534 < 1 .Figure 3 provides a clear illustration of this pattern.Also in Fig. 4, a simple calculation shows that the deterministic system given by Eq. ( 2) meets the requirement when we select β 1 = 0.75 and β 2 = 0.75 while leaving all other parameter values constant.This scenario is similar to the one shown in Fig. 3.
< σ 2 and R0 = 1.5 > 1 , Accordingly, Theorem 5 states that the infected individuals in the vector and human populations are, in fact, present in the system given by Eq.
(2); Fig. 4 illustrates this reality.The Fig. 5, graph shows a comparison between the stochastic and deterministic human systems with the p a r a m e t e r v a l u e s s e t a s f o l l o w s : When these parameter values are applied, a simple calculation shows that in the deterministic system.Eq. ( 1) R 0 = 0.75 < 1 and Eq.(2)σ 2 < max In Fig. 6, we assigning the following value to parameters, to compared the behavior of deterministic and stochastic model for both humans and vectors separately in this figure where, In Fig. 7, the comparative impact of prevention control parameter on disease transmission can be observed in Fig. 7.We can clearly see from the comparison that disease approaches almost zero for the maximum value of while keeping other values constant.
Figure 8 shows the impact of treatment control parameter on disease transmission.This can be observed that has comparatively lesser effect on disease transmission for its maximum value.While keeping other values constant.
In Fig. 9, we compare the impact of insecticide control parameter on disease transmission for its maximum and minimum values.The fact can be cleanly observed that maximizing has a greater impact on disease control by changing the value of from minimum to maximum.
In Fig. 10, the following parameter values, the effect of maximum values of control parameters are shown in Fig. 10: With the combine application of all control parameters, disease vanishes quickly.

Ethics approval
Our study did not require an ethical board approval because it did not contain human or animal trials.

Conclusion
The computational investigation of the stochastic ZIKV optimal control model using the LSCM determined several key vitality.The stability, convergence and accuracy properties of the proposed LSCM make it a vigorous choice for solving such complex differential equations inherent in a epidemiological systems.The stability analysis confirms that given LSCM maintains the probity of the approximate solution over time by ensuring that the errors do not grow up exponentially.Similarly the convergence analysis further highlights the efficacy of the proposed method which shows the spectral coefficients that rapidly decay, ensuring the reliable convergence to the true solution.However, the accuracy of the LSCM is evident through its exponential convergence which grant for efficient approximation of functions with having minimal computational effort.
Moreover, to incorporating the random interference into the dynamical model through white noise improve the reality of the simulations that the stochastic nature reflecting the disease transmission.The choice of parameter values, totaly based on a epidemiological data and realistic biological phenomena that ensures the simulations are both meaningful and relevant.The given approach provides beneficial understanding into the optimal control strategies for executing the spread of the ZIKV which show the practical applications of advanced numerical techniques in epidemiology and public health.
Overall, the LSCM proves to be a powerful tool for the computational modeling such as stochastic epidemiological models which offering the high stability, convergence rates and accuracy.The LSCM application to

Figure 7 . 9 Figure 8 . 1 Figure 9 .Figure 10 .
Figure 7. Different values for the prevention control parameter µ 1 ; as we increase then both the disease classes goes to zero.